Prove $\overline{X_n}\overset{P}{\rightarrow}c$ for some $c\in \mathbb{R}$ if $(X_n)$ is iid and $\phi(t)=\frac{1-e^{-2it}}{2it}$.

Question

Let $(X_n)$ be a sequence of iid random variables, whose common characteristic function is $\phi(t)=\frac{1-e^{-2it}}{2it}$ for $t\neq 0$. Prove that $\overline{X_n}\overset{P}{\rightarrow}c$ (convergence in probability) for some $c\in \mathbb{R}$.

Attempt. According to the SLLN, if $\mu=\mathbb{E}(X_1)\in \mathbb{R}$,then $\overline{X_n}\overset{a.s.}{\rightarrow}\mu$ and therefore $\overline{X_n}\overset{P}{\rightarrow}\mu$. So it would be enough to prove that the mean exists in $\mathbb{R}.$ But the formula $\phi'(0)=i\mathbb{E}(X)$ holds as long as we know that the mean value exists (in that case the mean would be equal to $1$).

Another approach would use convergence in distribution to some constant $c$ (possibly $c=1$): $$\phi_{\overline{X_n}}(t)=\left(\frac{1-e^{-2i\frac{t}{n}}}{2i\frac{t}{n}}\right)^n \to e^{ict},~~n\to +\infty,$$ but the limit seems hard to evaluate.

Thanks in advance.

Answer 1

The approach using the law of large numbers is fine. You have a small typo: the formula should be $\phi'(0) = -i \mathbb{E}[X]$ so the mean is $-1$. (You can actually show that $X_n \sim \text{Uniform}(-2, 0)$ for all $n$, but this is not necessary.)